Chain-of-thought (CoT) monitors are LLM-based systems that analyze reasoning traces to detect when outputs may exhibit attributes of interest, such as test-hacking behavior during code generation. In this paper, we use information-theoretic analysis to show that non-zero mutual information between CoT and output is a necessary but not sufficient condition for CoT monitorability. We identify two sources of approximation error that may undermine the performance of CoT monitors in practice: information gap, which measures the extent to which the monitor can extract the information available in CoT, and elicitation error, which measures the extent to which the monitor approximates the optimal monitoring function. Empirically, we show that CoT obfuscation can arise even in the absence of any optimization pressure against CoT. We further demonstrate that the fragility of CoT monitorability can be systematically mitigated through targeted training objectives. To this end, we propose a practical, label-free objective based on mutual information between outputs and CoTs, and benchmark it against an oracle-based method that directly rewards the monitored model for CoTs that maximize monitor accuracy. Across multiple environments, we show that our objective is effective at preventing CoT obfuscation even when training against a monitor, and helps mitigate reward hacking when the task reward is imperfectly specified.

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